The Science Of Rating And Ranking Who’s #1, by Amy N. Langville & Carl D. Meyer

Truth be told, this is not about the science of rating and ranking as much as it is about the mathematics of it, but since saying something is scientific is way cooler than saying it is based on linear algebra and probability theory, the authors of this book can be forgiven for pandering to popular tastes even while writing a book that will surely appeal mainly to the math nerds and sports nerds, and those who happens to be nerds about math, sports, politics, gambling, and a host of other related fields. Where this book excels is in demonstrating that there are many good algebraic methods for determining rank, that none of them is perfect, that there are always tradeoffs, and that the ability to customize and weight factors means that one can literally run wild in the hopes of creating worthwhile and potentially profitable statistical analysis, given the examples of fame and publicity that the book talks about, along with some people (most notably an insanely prolific Catalan author named Ramon Llull) who are undeserved obscure. The book’s wisest advice to those who wish to develop their own ranking methods for sports or other situations is that those who come up with the math do not receive nearly the fame and fortune as those who find a profitable and worthwhile use for the mathematical technique.

In terms of its contents, this book is very well-organized and practical, and also manages to make learning linear algebra with practical implications fun, and full of intriguing historical tips. After introducing with some unusual implications of Arrow’s Theroem (which says it is impossible to get everything you want from a ranking), the book goes through a selection of practical methods for ranking, from Massey’s to Colley’s to Keener’s (which prompts discussion on skew and normalization), to the popular ELO system used in chess and soccer, to Markov methods (a sort of random walk), to a football-appropriate Offense-Defense rating method. After this comes a discussion on reordering methods, point spreads (for the would-be gamesters among us), user preference ratings (for movies on Netflix, for example), handling ties (including inducing ties in certain circumstances, which improves the predictive value of many ranking systems), incorporating weights (like the ELO does for soccer by weighting friendlies far lower than World Cup games, for example), and looking at the sensitivity of rating systems to first-order “what if” scenarios. The book closes with a couple of chapters on rank aggregation (which would have been useful when I was working on my own music charts as a teen) , methods of comparison between two ranking lists, the messiness of data, and an epilogue that briefly discusses many of the methods that were left out to keep the book at a manageable length, including “big data” statistical analyses.

If you understood even half of what I was saying in the previous paragraph, the odds are that you would appreciate this book. The authors make a note that the ideal reader will have taken some coursework in Linear Algebra, and have a basic familiarity with concepts like eigenvectors and matrices, as they are covered in great detail in many of the methods. Even without a sophisticated understanding of the mathematics involved in optimizing ranking systems, the book is worthwhile for encouraging a practical understanding of how to determine what is the best. Given that many of us at least informally have to make decisions between alternatives, it is good to be able to have some sort of conceptual understanding of how to engage in that task better. As long as evaluation is a task that we have to deal with, knowing where to place people, books, sports teams, songs, insurance plans, or any other number of matters will be an area of importance. This book, in providing a history of ranking methods, some intriguing stories of the people involved in creating them and their purposes, and in providing some practical assistance in methodology, do a great service to those willing to read and reflect on the material. This is a book that ought to make the mathematics departments of College of Charleston and North Carolina State proud.